On the number of gaps of sequences with Poissonian pair correlations

A sequence (x _n) on the unit interval is said to have Poissonian pair correlation if #{1≤i≠j≤N:‖x _i−x _j‖≤s/N}=2sN(1+o(1)) for all reals s>0, as N→∞. It is known that, if (x _n) has Poissonian pair correlations, then the number g(n) of different gap lengths between neighboring elements of {x ₁,…,x _n} cannot be bounded along any index subsequence (n _t). First, we improve this by showing that, if (x _n) has Poissonian pair correlations, then the maximum among the multiplicities of the neighboring gap lengths of {x ₁,…,x _n} is o(n), as n→∞. Furthermore, we show that for every function f:N ⁺→N ⁺ with lim _n⁡f(n)=∞ there exists a sequence (x _n) with Poissonian pair correlations such that g(n)≤f(n) for all sufficiently large n. This answers negatively a question posed by G. Larcher.